Stochastic demand as an addition to the Solow Growth Model

The subject of this research is the Solow Growth Model. The relevance is substantiated by the fact that the Solow Growth Model is conceptually simple, and simultaneously it can be complicated with clarifications and additions. The authors believe that one of such clarification is consideration of the demand as a stochastic variable. The goal of this research is to propose a model that takes into account the random nature of consumer demand based on the Solow Growth Model. The article aims to examine the Solow Growth model; conduct a literature overview of the most common modifications of the model; analyze the well-known modifications and complications of the model; outline the methods of such modifications and complications; offer Solow Growth Model supplemented with microeconomic substantiation with consideration of the stochastic demand. The article employs the methods of analysis, synthesis, comparison, and differential calculus. The novelty lies in the statement  that consumption depends on demand; it is intuitively obvious that demand can be considered as stochastic variable. This is explained by the individual traits of the consumers. Therefore, the demand can be described via stochastic differential equation based on the standard Wiener process (analogy with Brownian motion). The article offers a stochastic differential equation of demand. The Solow Growth Model is supplemented with the stochastic differential equation of demand. In conclusion, the authors determine the key modification and complication trends of the Solow Growth Model; developed the model based on the Solow Growth Model with the stochastic differential equation of demand as its addition. Further research should be aimed at solution of the obtained mathematical model supplemented with the stochastic differential equation of demand.

Individual Account Pension Actuarial Model under Fixed Interest Rates

In the context of the global population aging, pension issue has become highly important and pressing national concerns. Therefore, to further improve and perfect pension system, especially for the pension insurance system, has a very important social value. From the following aspects of old-age insurance personal accounts actuarial model: The first, deduced fixed interest rate, monthly payment of personal accounts pension actuarial model; the second, taking into account the impact of different factors on interest rates, constructed in the form of joint modeling standard Wiener process and Poisson process personal accounts pension actuarial model formula is derived.

Delay-Dependent Stochastic Stability Analysis of Singular Hybrid System

The problem of Stochastic stability analysis for singular time-delay hybrid system with Markovian jumping parameters and standard Wiener process in this paper. Based on linear matrix inequality approach, a delay dependent condition is proposed, which ensures the singular hybrid system is stochastically stable.

The Expected Discounted Penalty Function under Stochastic Discount Interest Force Driven by Markov Process

We investigate the expected discounted penalty function in which the discount interest process is driven by markov process. We obtain the integro-differential equation satisfied by the expected discounted penalty function when interest process is perturbed by standard Wiener process and Poisson-Geometric process. A system of Laplace transforms of the expected discounted penalty function, given the initial environment state, is established from a system of integro-differential equations. One example is given with claim sizes that have exponential distributions.

A Limit Theorem for Random Products of Trimmed Sums of i.i.d. Random Variables

Let be a sequence of independent and identically distributed positive random variables with a continuous distribution function , and has a medium tail. Denote and , where , , and is a fixed constant. Under some suitable conditions, we show that , as , where is the trimmed sum and is a standard Wiener process.

The Markov Risk Model under Stochastic Discount Interest Force

We investigate the expected discounted penalty function in which the discount interest process is driven by markov process. We obtain the integro-differential equation satisfied by the expected discounted penalty function when interest process is perturbed by standard Wiener process and Poisson-Geometric process. A system of Laplace transforms of the expected discounted penalty function, given the initial environment state, is established from a system of integro-differential equations. One example is given with claim sizes that have exponential distributions.

On the Expected Discounted Penalty Function for a Risk Model with Thinning Process

This paper studies the expected discounted penalty function for a risk model in which the arrival of insurance policies is a Poisson process and the process of claim occurring is -thinning process. Using backward differential argument, we derive the integro-differential equation satisfied by the expected discounted penalty function when the stochastic discount interest process is perturbed by standard Wiener process and Poisson-Geometric process. Applications of the integral equation are given to the Laplace transform of the time of ruin, the deficit at ruin, the surplus immediately before ruin occurs. In some special cases with exponential distributions, closed form expressions for these quantities are obtained.

The Gerber-Shiu Discounted Penalty Function for Risk Process with Double Markovian Environment

In this paper, we study the Gerber-Shiu discounted penalty function. We shall consider the case where the discount interest process and the occurrence of the claims are driven by two distinguished Markov process, respectively. Moreover, in this model we also consider the influence of a premium rate which varies with the level of free reserves. Using backward differential argument, we derive the integral equation satisfied by the expected discounted penalty function via differential argument when interest process in every state is perturbed by standard Wiener process and Poisson process. In some special cases, closed form expression for these quantities are obtained.

BIFURCATIONS IN APPROXIMATE SOLUTIONS OF STOCHASTIC DELAY DIFFERENTIAL EQUATIONS

We consider stochastic delay differential equations of the form [Formula: see text] interpreted in the Itô sense, with Y(t)=Φ(t) for t∈[t0-τ,t0] (here, W(t) is a standard Wiener process and τ>0 is the constant "lag", or "time-lag"). We are interested in bifurcations (that is, changes in the qualitative behavior of solutions of these equations) and we draw on insights from the related deterministic delay differential equation, for which there is a substantial body of known theory, and numerical results that enable us to discuss where changes occur in the behavior of the (exact and approximate) solutions of the equation. Rather diverse components of mathematical background are necessary to understand the questions of interest. In this paper we first review some deterministic results and some basic elements of the stochastic analysis that (i) suggests lines of investigation for the stochastic case and (ii) are expected to facilitate the theoretical investigation of the stochastic problem. We then present the results of numerical experiments that illustrate some of the complexities that arise when considering bifurcations in stochastic delay differential equations. They give prima facie evidence for certain convergence properties of the bifurcation points estimated using the Euler–Maruyama method for the equations considered. We conclude by drawing attention to a number of open questions in the field.

Rezimi kursnog koridora u kontekstu valutnih kriza

One of the pillars of open-economy macro is the well-known axiom of "inconsistent trinity", which claims it is impossible to simultaneously achieve significant liberalisation of international capital movements, monetary policy independence and fixed exchange rate. Ignorance or inability to accept this identity, have recently placed many developing countries with fixed (yet adjustable) exchange rate regimes in the centre of fierce international financial crises with both monetary and real repercussions. Precautionary or after being burnt by crisis, majority of small open economies adopted the so-called bipolar view, i.e. one of exchange rate modalities which stretch along the wide axis between rigid fixation and free floating. After having dealt with elementary dynamics of crawling/monitoring band intermediate regime, this paper analyses its behaviour in the face of speculative attacks against domestic currency. In doing so, first, we reminded that exchange rate trajectory, even inside ("honeymoon"-free) currency band, does not follow Brownian motion (standard Wiener process). We showed, thereafter, that imperfectly-credible target zones could also have an asymptotically stabilizing effect on intramarginal exchange rate dynamics. This mean-reverting trait enables monetary authorities to control speculative volatility of exchange rate even in cases when they lack sufficient amount of foreign reserves, inasmuch as fundamental or globally spread crises need not be defended by means of open-market intervention, but by unanticipated, temporally irregular inside-the-band realignment or simply by sustainable widening of the band. Nevertheless, in principle, currency band should be shielded against non-fundamental market pressures. Therefore, we felt it was worthwhile calibrating the imminent danger of initial and cumulative attacks cum total currency crash, by means of introducing formal mathematical relationships among key variables which reflect both structural and behavioural features of the economy at hand (and, primus inter pares, its exchange rate regime), on the one hand, and on the other hand, objective functions of both policymaker and speculators (individually). At first armed with ad hoc log-nonlinear stochastic model of currency band, and later on by intertemporal optimisation modeling, we reconstructed preattack exchange rate trajectory, "sufficient" amount of foreign exchange reserves, exchange rate behaviour during the currency crisis and sequence of crisis events. At last, we carried out rigorous analysis of variables which influence width and/or location of ranges of effective commitment. Finally, we suggested alleys which may prove fruitful for further research efforts.